Here's a counterexample in $n\ge3$ dimensions. With $\lVert\cdot\rVert$ denoting the Euclidean norm, set $$ f(x)=\begin{cases} x_1x_2x_3\lVert x\rVert^{-n-4},&{\rm if\ }x\not=0,\cr 0,&{\rm if\ }x=0. \end{cases} $$ This is harmonic on $x\not=0$, has first and second order derivatives at the origin, but $f$ along with all its derivatives are discontinuous at the origin.
Note that, $f=c\frac{\partial^3(\lVert x\rVert^{-n+2})}{\partial x_1\partial x_2\partial x_3}$ is a third order derivative of a harmonic function, so is harmonic (away from the origin). As $f$ vanishes whenever all but two coordinates are zero, it along with its first order derivatives all vanish on the coordinate axes. So, $f$ along with all its first and second order derivatives vanish at the origin.